Coexistence, Oscillations, and Chaos in a Discrete Predator-Prey System with Fear-Modulated Prey Growth

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Predator–prey interactions are governed not only by direct consumption but also by behavioral responses of prey to the perceived risk of predation. In this work, we formulate a discrete-time predator–prey model in which predator-induced fear modifies prey reproduction through an exponential suppression mechanism, while predator consumption follows a Holling type-II functional response. The resulting map combines Ricker-type prey growth, density-dependent predation, and a non-consumptive effect associated with predator presence. We establish fundamental dynamical properties of the system by proving positivity, boundedness, and persistence under appropriate parameter restrictions. The existence of biologically meaningful equilibria is determined, and their local behavior is characterized using the Jacobian matrix and Jury stability criteria. To examine the influence of ecological parameters on the dynamics, we employ a global sensitivity analysis based on the Partial Rank Correlation Coefficient (PRCC) approach. The analysis identifies the parameters that most strongly affect prey and predator abundance. Numerical investigations reveal a wide range of dynamical regimes, including stable coexistence, periodic oscillations, higher-period attractors, quasi-periodic motion, and chaos. In particular, flip and Neimark–Sacker bifurcations are observed as ecological parameters vary. Basin-of-attraction computations and two-parameter iso-spike diagrams further demonstrate the presence of multistability and strong dependence on parameter combinations and initial conditions. Finally, period-doubling control and pole-placement techniques are employed to suppress undesirable chaotic oscillations and recover stable coexistence. The results emphasize that predator-induced fear can substantially modify population fluctuations and may act as an important mechanism regulating coexistence and complex dynamics in discrete ecological systems.

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